# Publications of the Research Institute for Mathematical Sciences

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**Volume 49, Issue 2, 2013, pp. 271–312**

**DOI: 10.4171/PRIMS/106**

Published online: 2013-06-05

Matrix-valued Orthogonal Polynomials Related to (SU(2)$ \times$ SU(2), diag), II

Erik Koelink^{[1]}, Maarten van Pruijssen

^{[2]}and Pablo Román

^{[3]}(1) Radboud Universiteit Nijmegen, Netherlands

(2) Radboud Universiteit Nijmegen, Netherlands

(3) Universidad Nacional de Córdoba, Argentina

In a previous paper we have introduced matrix-valued analogues of the Chebyshev polynomials by studying matrix-valued spherical functions on SU(2)$\times$ SU(2). In particular the matrix-size of the polynomials is arbitrarily large. The matrix-valued orthogonal polynomials and the corresponding weight function are studied. In particular, we calculate the LDU-decomposition of the weight where the matrix entries of $L$ are given in terms of Gegenbauer polynomials. The monic matrix-valued orthogonal polynomials $P_n$ are expressed in terms of Tirao's matrix-valued hypergeometric function using the matrix-valued differential operator of first and second order to which the $P_n$'s are eigenfunctions. From this result we obtain an explicit formula for coefficients in the three-term recurrence relation satisfied by the polynomials $P_n$. These differential operators are also crucial in expressing the matrix entries of $P_nL$ as a product of a Racah and a Gegenbauer polynomial. We also present a group theoretic derivation of the matrix-valued differential operators by considering the Casimir operators corresponding to SU(2)$\times$ SU(2).

*Keywords: *matrix-valued orthogonal polynomials, spherical functions, matrix-valued differential operators, orthogonal polynomials, SU(2)

Koelink Erik, van Pruijssen Maarten, Román Pablo: Matrix-valued Orthogonal Polynomials Related to (SU(2)$ \times$ SU(2), diag), II. *Publ. Res. Inst. Math. Sci.* 49 (2013), 271-312. doi: 10.4171/PRIMS/106